The product of the lengths of the perpendiculars drawn from the foci to any tangent to the hyperbola $x^{2} - \frac{y^{2}}{4} = 1$ is:

The equation of the tangent at the point $(a \sec \theta, b \tan \theta)$ of the conic $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is:

The equation of the tangent parallel to $y - x + 5 = 0$ drawn to the hyperbola $\frac{x^2}{3} - \frac{y^2}{2} = 1$ is

If $\frac{x}{a} + \frac{y}{b} = 1$ is a tangent to the curve $x = Kt, y = \frac{K}{t}, K > 0$,then

$A$ tangent to the hyperbola $\frac{x^2}{4} - \frac{y^2}{2} = 1$ meets the $x-$axis at $P$ and the $y-$axis at $Q$. Lines $PR$ and $QR$ are drawn such that $OPRQ$ is a rectangle (where $O$ is the origin). Then $R$ lies on

The number of possible tangents which can be drawn to the curve $4x^2 - 9y^2 = 36$,which are perpendicular to the straight line $5x + 2y - 10 = 0$ is